of misspecied logarithmic random walks 1
by
Walter Kramer and Laurie Davies
Fachbereich Statistik, UniversitatDortmund, D-44221Dortmund
Fachbereich Mathematikund Informatik,Universitat Essen, D-45117Essen
Germany
1 Introduction and Summary
Testing for unit roots has been among the most heavily researched topics
in Econometrics for the last quarter of a century. Much less researched is
the equally important issue of the appropriate transformation (if any) of the
variableof interest whichshould preceed any such testing.
In macroeconometrics and empirical nance (stock prices, exchange rates),
thereare oftencompellingargumentsinfavorofalogarithmictransformation.
Elsewhere, forinstance inthe modellingof interestrates, alevelsspecication
automatically suggests itself.In many applications, however, it isnot a priori
clear, given that one suspects a unit root,whether this unit rootis present in
the levelsorthe logs, sothere iscertainly someinterest inthe testing forunit
roots in the context of an incompletely specied nonlinear transformation of
the data.
Thisissuecanbeapproachedfromvariousangles.Oneistocheckwhichtrans-
formations leave the I(1){property of a time series intact, the presumption
being that any such transformation could then do little damage to the null
distribution of atest forunit roots(Granger and Hallmann 1991,Ermini and
Granger 1993, Corradi 1995). A related one is to use tests whose null dis-
1
ResearchsupportedbyDeutscheForschungsgemeinschaftviaSFB475
data areI(1)ornot(Granger and Hallmann1991,Burridge andGuerre 1996,
Gourierouxand Breitung 1999), orto embed the levelsand logspecications,
respectively, inageneralBox-Cox-frameworkand toestimatethe transforma-
tion parameter before testing (Franses and McAleer 1998, Franses and Koop
1998, Kobayashi and McAleer 1999).
ThepresentpapercontinuesalongthelinesofGrangerandHallmann(1991)by
focussing ona conventional test procedure |the standard Dickey-Fuller-test
| and by investigating its properties under a misspecied nonlinear trans-
formation (in particular: investigating whether an existing unit root is still
detected, i.e. the null hypothesis of anexisting unit rootis not rejected when
aninappropriatetransformationisapplied).Giventhatthis test iswidelyem-
ployed,andgiventhatthechoicebetweenalinearandaloglinearspecication
is often rather haphazardly done in applications, it is important to know the
degree towhich the acceptance ofthe nullhypothesis depends onthe correct-
ness ofthe data transformation.
Granger and Hallmann (1991) nd through Monte Carlo that the standard
Dickey-Fuller-test overrejects a correct null hypotheses of a random walk in
the logs, when the test is instead applied to the levels. Below we prove ana-
lytically that the rejection probability can take arbitrary values between zero
and one for any sample size. An analogous result obtains when the levels fol-
low a random walk, but the Dickey-Fuller-test is applied to the logs. Again,
the rejection probability is shown to be depend on both the sample size and
the innovation variance, so the null distribution of the DF-test is extremely
non-robust to improperdata transformations.
test as applied to levels when the random
walk is in the logs
Thissectionisconcernedwiththe casewherethe truedatageneratingprocess
(DGP) is a random walk in the logs, but the Dickey-Fuller-test is inadver-
tantly appliedto the levelsof the data (i.e. to a multiplicativerandom walk).
The common wisdom, as pronounced by Granger and Hallmann (1991) or
..., isthat thatthe nulldistributionofthe DF-testis"mo-
re spread out", inducingoverrejections of the nullhypothesis. Below we show
that this is onlypartially true, as the rejection probability under H
0
depends
cruciallyontheinterplaybetween theinnovationvarianceand thesamplesize.
Let y
t
bethe time seriesunder study, letz
t
:=ln(y
t
), and assume that
z
t
=+z
t 1 +"
t
; "
t
nid(0;
2
); t=1;:::;T; z
0
= constant :(1)
The standardDickey-Fuller-test, asappliedtothe levelsy
t
:=exp(z
t
),is given
by T(^ 1),where
^ =
P
T
t=1 y
t 1 y
t
P
T
t=1 y
2
t 1
(2)
is the OLS-estimatorfor in the model
y
t
=y
t y +u
t
: (3)
Itassumesthatthere isnodriftcomponentinthe data.Itsasymptoticcritical
values,whichcanby nowbefound inmany textbooks,arec= 5;6(=1%),
c = 7;9( = 5%) and c = 13;3( = 10%). They are based on the null
hypothesis that in eq. (3), =1 and u
t
iid(0;
2
) (which, of course, is not
quite correctif the true DGP isgiven by (1)).
The objects of our study are the true rejection probabilities under H
0 , i.e.
P(T(^ 1) < c ). As the joint distribution of (z
1
;:::;z
T
) and therefore also
(y
1
;:::;y
t
) is uniquely determined by (given the distribution of the "'s),
these probabilities are functions of 2
and T. We consider the limits of these
probabilities for 2
!0 and 2
!1(T xed) and for T !1( 2
xed).
To determine the limiting probabilities for a given sample size as 2
! 0 or
2
!1 werst consider the limiting behaviourof .^
Theorem 1 If the data are generated by (1), and ^is the OLS-estimator for
in (2), we have:
a) 2
!0 implies ^ d
!1.
b) 2
! 1 implies that ^ has a limiting distribution with mass
P(max
t=0;:::;T 1 y
2
t
> y
T y
T 1
) at zero and mass P(max
t=0;:::;T 1 y
2
t
<
y
T y
T 1
) at innity.
The proof of the theorem is in the appendix. It does not make any use of the
iid{propertyoftheinnovationsandholdsforquitearbitraryjointdistributions
of ("
1
;:::;"
T ).
Theorem 1 immediately gives the limiting rejection probability for the stan-
dard DW-test, for one sided-tests and arbitrary critical values less than zero:
Rejection probabilities tend to zero as the innovation variancebecomessmall
(asT(^ 1)!0),and they tendtoP(max
t=0;:::;T 1 y
2
t
>y
T y
T 1
) astheinno-
vationvariance increases (assuming that the criticalvalue is largerthan T),
since T(^ 1)!1, whenevermaxy 2
t
>y
T y
T 1 .
The probability for the latter event depends only on the sample size T. It
is easily seen to tend to the probability that the maximum absolute value of
the random walk fz
t
g exceeds its nal value z
T
, which in turn converges to
...
Given T, the true rejection probabilities approach their limits (as ! 1 )
frombelow; they areanincreasing functionof,asshown inour MonteCarlo
experimentsin section5. This is intuitively obvious, since ...
The next theorem gives the limiting rejection probabilitiesas T !1 and 2
xed.
3 The null distribution of the Dickey-Fuller-
test as applied to logs when the random
walk is in the levels
This section considers the case where the true DGP is given by (1), and the
timeseriestowhichtheDickey{Fullertestisappliedisgiven y
t
:=ln(z
t
).This
type of misspecication appears to be less serious in practice, if onlybecause
ofpossiblenegativevalues ofz
t
.It canthereforeoccuronlyifz
0
isratherlarge
and/or a sizable drift component (to be considered later) prevents this from
happening.
The next theorem therefore assumes that z
t
> 0 ( t= 0 ;:::;T); it considers
the limiting behaviourof
^ (c):=
ln(cz
t 1 )ln(cz
t )
[ln(cz
t 1 )]
2
(4)
as c!0 and as c!1.
Theorem 2 If the data are generated by (1), we have (c)^ d
! 1, both as
c!0 and as c!1.
The proof is inthe appendix.
Theorem 1 implies that the conditional rejection probability of the DW-test
(giventhat ...)tendstozero whenthe DW-testisinadvertantlyappliedtothe
logs of a random walk. It possibly explains why Nelson and Plosser (1982),
in their seminal paper onunit roots in US{macroeconomic time series, found
unit roots inlog{transformed time series although, as argued by Franses and
McAleer (1998,p.160), the "true"unit rootis inthe levelsratherthan inthe
logs: Ifthe levelsseries hasa unitroot,this unit rootmighteven appearmore
signicant if the test is incorrectly applied tologs instead and the innovation
varianceis extremeenough.
drift component in (1) is 0, z
t
will with probability 1 eventually become
negative,sothereisnopointininvestigatingthelimitingconditionalrejection
probability when the conditioning event has vanishing probability. If > 0,
...
4 Some nite sample Monte Carlo evidence
Burridge, P.; Guerre E.: "Thelimitdistributionoflevelcrossingsofaran-
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705 { 723.
Corradi, V.; Swanson, N.: "Choosing between levels and logs in the pre-
senceofdeterministicandstochastictrends."Mimeo,UniversityofPenn-
sylvania1997.
Corradi, V.: "Nonlineartransformationsofintegratedtimeseries: areconsi-
deration." Journal of Time Series Analysis16,1995, 539 { 549.
Ermini, L.; Granger, C.W.J.: "Some generalizations on the algebra of
I(1)-processes." Journal of Econometrics 58, 1993,369 {384.
Ermini, L.; Hendry, D.F.: "Log income versus linear income: An applica-
tion of the encompassing principle." Mimeo, Nueld College, Oxford
1995.
Franses, P.H.; Koop, G.: "On thesensitivityofunit rootinferencetonon-
linear data transformations."Economics Letters 59, 1998, 7{ 15.
Franses, P.H.; McAleer, M.: "Testing forunit rootsand non-linear trans-
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Granger, C.W.J.; Hallmann, J.: "Nonlinear transformations of integra-
ted time series." Journal of Time Series Analysis12, 1991, 207 { 224.
Guerre, F.; Jouneau, F.: "Geometric versus arithmetic randomwalk: The
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Nelson, C.R.; Plosser, C.I.: "Trendsand randomwalksinmacroeconomic
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Proof of Theorem 1:
Consider the distributionof
^ =
P
T
t=1 y
t 1 y
t
P
T
t=1 y
2
t 1
(A.1)
for some given >0. The distributionof ,^ when is replaced by ~:=c, is
then identical tothe distributionof
^ (c):=
P
T
t=1 (y
t 1 y
t )
c
P
T
t=1 (y
2
t 1 )
c
: (A.2)
As both the numerator and the denominator in (A1) tend to T as c ! 1 ,
part(a) of the theorem follows.
To determine the limiting distributionof (c)^ as c!1 ,keepy
1 :::;y
T xed
and let y 2
k 1
=max
t=1;:::;T y
2
t 1
. Then
^ (c)=
T
P
t=1 y
t 1 y
t
y 2
k 1
!
c
T
P
t=1 y
2
t 1
y 2
k 1
!
c
; (A.3)
and this expression can have only two limits as c ! 1 (assuming without
loss of generality that y 2
k 1 6= y
T y
T 1 ). If y
2
k 1
> y
T y
T 1
, then y 2
k 1
> y
t y
t 1
for all t = 1 ;:::;T, so the numerator tends to zero as c ! 1 . Since the
denominator tends to unity (neglecting the possibility that k is not unique),
we have (c)^ !0.
If y 2
k 1
< y
T y
T 1
, the denominator still tends to unity, but the numerator
tends innity,so (c)^ !1.This impliesthat the limitingdistributionof(c)^
as c!1 is degenerate, with mass P(y 2
k 1
>y
T y
T 1
) atzero, and the rest at
innity (inthe sense that, for allm 2IN, >0, and ">0there is ann such
that
P (0(c)^ <"or (c)^ >m)>1 " for alln>n
: (A.4)
Proof of Theorem 3:
Wehave
^ (c) =
ln(cz
t 1 )ln(cz
t )
[ln(cz
t 1 )]
2
=
lnz
t 1 lnz
t
+lnc(lnz
t 1 +lnz
t
)+Tln(c) 2
(lnz
t 1 )
2
+2ln(c)ln(z
t 1
)+Tln(c) 2
:
and this expression tends to1 both asc!0and c!1.
